Sperner's Lemma

In this article we introduce and prove properties of simplicial complexes in real linear spaces which are necessary to formulate Sperner's lemma. The lemma states that for a function f , which for an arbitrary vertex v of the barycentric subdivision B of simplex K assigns some vertex from a face of K which contains v, we can find a simplex S of B which satisfies f (S) = K (see [10]). The notation and terminology used in this paper have been introduced in the We follow the rules: x, y, X denote sets and n, k denote natural numbers. The following two propositions are true: (1) Let R be a binary relation and C be a cardinal number. If for every x such that x ∈ X holds Card(R • x) = C, then Card R = Card(R(dom R \ X)) + C · Card X. (2) Let Y be a non empty finite set. Suppose Card X = Y + 1. Let f be a function from X into Y. Suppose f is onto. Then there exists y such that y ∈ Y and Card(f −1 ({y})) = 2 and for every x such that x ∈ Y and x = y holds Card(f −1 ({x})) = 1.